Thermodynamic Properties of Ternary Aqueous Solutions with the

properties of the ternary aqueous systems nitrate, chloride, or sulfate with magnesium, {Mg/Cl/NO3/SO4}(aq), are compared with the data obtained for t...
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2188

J. Chem. Eng. Data 2007, 52, 2188-2194

Thermodynamic Properties of Ternary Aqueous Solutions with the Common Magnesium Cation {Mg/Cl/NO3/SO4}(aq) at T ) 298.15 K Mohamed El Guendouzi* and Abdelkbir Errougui Laboratoire de Chimie Physique, De´partement de Chimie, Faculte´ des Sciences Ben M’sik, Universite´ Hassan II Mohammedia, B.P 7955, Casablanca, Morocco

In this investigation, ternary aqueous electrolytes of chloride, nitrate, and sulfate with the common magnesium cation have been studied using the hygrometric method at 298.15 K. The water activities of the ternary systems {yMgCl2 + (1 - y)MgSO4}(aq) and {yMg(NO3)2 + (1 - y)MgSO4}(aq) are measured at total molalities from 0.35 mol‚kg-1 to saturation for different ionic strength fractions (y) of MgCl2 or Mg(NO3)2, respectively, with y ) 0.20, 0.50, and 0.80. The obtained data allow the deduction of osmotic coefficients of the solvent. From these measurements, mixing ionic parameters are determined and used to predict the solute activity coefficients in the mixture, using Pitzer’s model. The parameters θMN and ψMNX are evaluated for these systems. The thermodynamic properties of the ternary aqueous systems nitrate, chloride, or sulfate with magnesium, {Mg/Cl/NO3/SO4}(aq), are compared with the data obtained for the system [MgCl2 + Mg(NO3)2](aq) already published in a previous work.4 The comparison shows that the hydration is affected by the nature of the solute, MgSO4, MgCl2, or Mg(NO3)2, in the mixed aqueous solutions. These effects might be influenced by ion pairing, with no contact in Mg(NO3)2(aq) and with ion association in MgSO4(aq).

1. Introduction Knowledge of thermodynamic properties for aqueous solutions with magnesium is of great interest from both theoretical and practical viewpoints. Aqueous solutions of Mg2+ combined with Cl-, SO42-, and NO3- are important components in many natural and industrial processes.1-3 Various chemical processes and industrial problems require thorough knowledge of the properties of aqueous solutions including those of the Mg2+ cation. The study of electrolytes is fundamentally important to understand the properties and behavior of aqueous multicomponent systems. Much attention has been paid to chloride, sulfate, nitrate, magnesium, and their mixtures. The mixed aqueous electrolytes with common cations or anions continue to be interesting. This paper presents research on ternary aqueous mixtures of magnesium chloride, sulfate, and nitrate, {Mg2+/Cl-/NO3-/SO42-}(aq), that are a continuation of our previous work on the ternary aqueous system [MgCl2 + Mg(NO3)2](aq).4 In the present work, the thermodynamic properties of the aqueous systems [MgCl2 + MgSO4](aq) and [Mg(NO3)2 + MgSO4](aq) were determined using the hygrometric method at 298.15 K. These systems have not been extensively studied in this way. Wu et al.5 measured the osmotic coefficients of the solvent of the system [MgCl2 + MgSO4](aq) by the isopiestic method at 298.15 K in the range of ionic strengths 0.8 to 6.8. Also, Ha and Chan6 evaluated the water activities of the systems [MgCl2 + MgSO4](aq) and [Mg(NO3)2 + MgSO4](aq) using an electrodynamic balance (EDB) method at ambient temperature 293.95 K, for the molar ratio 1:1 in the range of water activity from 0.80 to 0.42. We report the results of measurements of water activities for the total molality range from 0.35 mol‚kg-1 to saturation for mixed electrolytes {yMgCl2 + (1 - y)MgSO4}(aq) and {yMg* Corresponding author. E-mail: [email protected]. Fax: (212) (2) 70-46-75.

Figure 1. Description of materials used in the hygrometric apparatus: a, microscope; b, lid; c, support of the drops; d, cup; e, wooden box.

Figure 2. Variation of the ratio K of diameter droplets as a function of the water activity aw.

(NO3)2 + (1 - y)MgSO4}(aq) at different ionic strength fractions (y) of MgCl2 (y ) IMgCl2/(IMgCl2 + IMgSO4)) or Mg(NO3)2 (y ) IMg(NO3)2/(IMg(NO3)2 + IMgSO4)), respectively, with y ) 0.20, 0.50, and 0.80 at 298.15 K. The osmotic coefficients are evaluated for these solutions from the water activities. The experimental data are used for the calculation of solute activity coefficients using the Pitzer model with our ionic mixing parameters. A comparison of the behavior of aqueous electrolyte solutions with magnesium is reported and shows that the hydration in these solutions is affected by the nature of the solute, MgSO4, MgCl2, or Mg(NO3)2.

10.1021/je7002176 CCC: $37.00 © 2007 American Chemical Society Published on Web 10/03/2007

Journal of Chemical and Engineering Data, Vol. 52, No. 6, 2007 2189 Table 1. Ratios of Growth K of the NaCl(aq) Droplets, Water Activities aw, and Osmotic Coefficients O of the Solvent to the Mixture {yMgCl2 + (1 - y)MgSO4}(aq) at Total Molalities mtot for Different Ionic Strength Fractions (y) of MgCl2a mMgCl2

MMgSO4

mtot

mol‚kg-1

mol‚kg-1

mol‚kg-1

0.15 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.74

0.45 0.60 0.75 0.90 1.20 1.50 1.80 2.10 2.22

0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0.80 1.07 1.30 1.60 1.85 2.13 2.40 2.78 3.20

Table 2. Ratios of Growth K of the NaCl(aq) Droplets, Water Activities aw, and Osmotic Coefficients O of the Solvent to the Mixture {yMg(NO3)2 + (1 - y)MgSO4}(aq) at Total Molalities mtot for Different Ionic Strength Fractions (y) of Mg(NO3)2a mMg(NO3)2

MMgSO4

mtot

K

aw

φ

mol‚kg-1

mol‚kg-1

mol‚kg-1

y ) 0.20 0.60 0.80 1.00 1.20 1.60 2.00 2.40 2.80 2.96

1.074b 1.802 1.642 1.532 1.356 1.231 1.126 1.056 1.030

0.9835 0.9772 0.9706 0.963 0.946 0.925 0.898 0.870 0.857

0.684 0.711 0.737 0.774 0.859 0.966 1.108 1.229 1.287

0.30 0.45 0.60 0.75 0.90 1.20 1.50 1.80 1.95 2.10

0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 1.35 1.50

y ) 0.50 0.35 0.70 1.05 1.40 1.75 2.10 2.45 2.80 3.15 3.50

0.10 0.15 0.20 0.25 0.30 0.40 0.50 0.60 0.65 0.70

1.166b 1.676 1.438 1.278 1.167 1.085 1.017 0.964 0.920 0.895c

0.9870 0.9719 0.955 0.934 0.910 0.882 0.850 0.816 0.777 0.738

0.807 0.879 0.951 1.059 1.165 1.293 1.435 1.571 1.726 1.876

0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60

0.15 0.20 0.24 0.30 0.35 0.40 0.45 0.52 0.60

y ) 0.80 0.95 1.27 1.54 1.90 2.20 2.53 2.85 3.30 3.80

0.80 1.05 1.30 1.60 1.85 2.16 2.40 2.64 2.90 3.20

1.393 1.252 1.151 1.063 1.003 0.951 1.909 0.877c 0.829

0.951 0.929 0.906 0.873 0.842 0.805 0.766 0.708 0.640

1.033 1.132 1.251 1.396 1.527 1.675 1.827 2.043 2.294

a The reference water activity is 0.84, and the reference solution is NaCl(aq). b Reference water activity is 0.98. c Reference solution is LiCl(aq).

K

aw

φ

y ) 0.20 0.4 0.6 0.8 1.0 1.2 1.6 2.0 2.4 2.6 2.8

1.234b 1.078b 1.872 1.666 1.544 1.372 1.246 1.149 1.106 1.072

0.9893 0.9837 0.9778 0.9714 0.964 0.948 0.928 0.905 0.891 0.877

0.662 0.675 0.691 0.716 0.749 0.832 0.928 1.032 1.095 1.152

0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20

y ) 0.50 0.35 0.70 1.05 1.40 1.75 2.10 2.45 2.80

1.180b 1.708 1.463 1.302 1.187 1.103 1.040 0.988

0.9875 0.9733 0.957 0.938 0.915 0.890 0.862 0.832

0.779 0.835 0.898 0.989 1.092 1.196 1.309 1.419

0.15 0.20 0.25 0.30 0.35 0.41 0.45 0.50 0.54 0.60

y ) 0.80 0.95 1.25 1.55 1.90 2.20 2.57 2.85 3.14 3.44 3.80

1.420 1.278 1.175 1.087 1.030 0.973 0.937 0.914c 0.890c 0.863c

0.953 0.934 0.912 0.884 0.857 0.822 0.793 0.763 0.729 0.688

0.990 1.068 1.162 1.267 1.371 1.491 1.589 1.683 1.794 1.922

a The reference water activity is 0.84, and the reference solution is NaCl(aq). b Reference water activity is 0.98. c Reference solution is LiCl(aq).

2. Experimental Section The water activity was determined using the hygrometric method described in a previous work.7 The apparatus used in this study for mixed electrolyte solutions is essentially the same as that used for the investigation of aqueous single electrolyte solutions. It is based on the measurement of the relative humidity over an aqueous solution containing nonvolatile electrolytes. The apparatus used is a hygrometer in which a droplet of salt solution is maintained on a thin thread. The diameter measurement of the previously calibrated droplet therefore permits knowledge of the relative humidity of the aqueous solution. The drops are composed of sodium or lithium chloride solutions. The droplets of a reference solution of NaCl(aq) or LiCl(aq) are deposited on the spider-thin thread by pulverisation. This thread is kept tense over a perspex support, which is fixed to a cup containing the selected studied solution (Figure 1). The cup is then placed in a thermostated box. The droplet diameter is measured by a microscope with an ocular equipped with a micrometric screw. The relative humidity is equivalent to the water activity aw in our experiments. From measurements of reference droplet diameters D(aw(ref)) above the reference solution and the same diameters D(aw) above the studied solution, we calculate the ratio of growth K (K ) (D(aw))/(D(aw(ref)))) and determine graphically the water activity using the variation of the ratio K (Figure 2) as a function of the water activity of reference solution NaCl(aq) or LiCl(aq).

Figure 3. Water activities aw of {yMgCl2 + (1 - y)MgSO4}(aq) at total molalities mtot, for different ionic strength fractions (y). O, y ) 0.00 (MgSO4 (aq)); 0, y ) 0.20; (, y ) 0.50; 2, y ) 0.80; +, y ) 1.00 (MgCl2(aq)).

Generally, the reference relative humidity is 0.84. For dilute solutions, the water activity reference is 0.98. For water activities of greater than 0.75, a solution of NaCl or LiCl is used, whereas for water activities of less than 0.75, a solution of LiCl is used. The uncertainty in the water activity depends on the precision of the diameter measurements and is less than ( 0.02 % for aw > 0.97, ( 0.05 % for aw > 0.95, ( 0.09 % for aw > 0.90, and ( 0.2 % for aw > 0.85. Also the overall uncertainty of the osmotic coefficient is estimated to be, at most, ( 6‚10-3.

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Journal of Chemical and Engineering Data, Vol. 52, No. 6, 2007

Figure 4. Water activities aw of {y Mg(NO3)2 + (1-y)MgSO4}(aq) at total molalities mtot, for different ionic-strength fractions (y). O, y ) 0.00 (MgSO4(aq)); 0, y ) 0.20; (, y ) 0.50; 2, y ) 0.80; +, y ) 1.00 (Mg(NO3)2(aq)).

Figure 5. Water activities aw of magnesium ternary systems at total molalities mtot at an ionic strength fraction of y ) 0.50. 0, MgCl2-MgSO4(aq); (, Mg(NO3)2-MgSO4(aq); 2, MgCl2-Mg(NO3)2(aq).4

The solutions of NaCl, LiCl, MgCl2, MgSO4, and Mg(NO3)2 were prepared from crystalline material (Extrapur-grade chemicals, NaCl 99.50 %, LiCl, MgCl2‚6H2O 99.00 %, Mg(NO3)2‚6H2O 99.00 %, and MgSO4 99.00 %) and deionized distilled water. The reference solutions are NaCl(aq) or LiCl(aq).

3. Results and Discussion 3.1 Water ActiWity and Osmotic Coefficient. In this work, the water activity was measured for the ternary systems{yMgCl2 + (1 - y)MgSO4}(aq) and {yMg(NO3)2 + (1 - y)MgSO4}(aq), at a total molality range from 0.35 mol‚kg-1 to saturation at 298.15 K. We report the results of measurements of water activities at different ionic strength fractions (y) of MgCl2 (y ) IMgCl2/(IMgCl2 + IMgSO4)) or of Mg(NO3)2 (y ) IMg(NO3)2/(IMg(NO3)2 + IMgSO4)), respectively, with y ) 0.20, 0.50, and 0.80. The experimental values of water activity are listed in Tables 1 and 2 and presented in Figures 3 and 4. The ranges of water activity are presented, for each ternary system: 0.9752 > aw > 0.504 for [MgCl2 + MgSO4](aq); 0.9906 > aw > 0.357 for [Mg(NO3)2 + MgSO4](aq); and we report also for [MgCl2 + Mg(NO3)2](aq),4 0.9906 > aw > 0.269.

Figure 6. Osmotic coefficient φ of water for the ternary systems with magnesium Mg2+/Cl-/NO3-/SO42- against the square root of total ionic strength I1/2 at different ionic strength fractions (y). O, y ) 0.00; 0, y ) 0.20; (, y ) 0.50; 2, y ) 0.80; +, y ) 1.00.

These figures show that the water activity decreases with increases of concentration. It is noted that the curves for the ternary systems are between those for the corresponding binary systems in the following order: aw(MgSO4(aq)) > aw(y ) 0.20) > aw(y ) 0.50) > aw(y ) 0.80) > aw(Mg(NO3)2) or aw(MgCl2(aq)). For comparison of the behavior of these systems with magnesium, we have presented in Figure 5 the water activity curves for the studied ternary systems Mg2+/Cl-/NO3-/SO42at an ionic strength fraction of y ) 0.50. The plots are practically identical for [MgCl2 + MgSO4](aq) and [Mg(NO3)2 + MgSO4](aq) in the studied range of molality, but the electrolyte [MgCl2 + Mg(NO3)2](aq) presents a lower water activity than the systems containing magnesium sulfate. The variation of water activities shows that the hydration of the magnesium electrolytes Mg2+/Cl-/NO3-/SO42- decreases in the following order: hyd(Cl/NO3) > hyd(Cl/SO4) ∼ hyd(NO3/SO4). This behavior was observed in binary aqueous solutions8 and is explained by the hydration phenomenon. The amount of water molecules required to hydrate the ions increases, and the amount of free solvent decreases. Therefore, the quantity of solvent which can pass into the vapor phase diminishes, and the most extensively hydrated salt results in a relatively low water activity. This order of hydration is influenced by the presence

Journal of Chemical and Engineering Data, Vol. 52, No. 6, 2007 2191 Table 3. Ionic Interaction Parameters θMN and ψMNX in Ternary Systems {Mg2+/Cl-/NO3-/SO42-}(aq) and Their Binary β(0), β(1), β(2), and CO at 298.15 K electrolyte

β(0)

β(1)

β(2)

MgCl2 Mg(NO3)2 MgSO4 ternary system with common cation {Mg2+/Cl-/NO3-/SO42-}(aq) MgCl2 + Mg(NO3)2 + H2O MgCl2 + MgSO4 + H2O

0.3492 0.3216 0.2305 θMN

1.6653 1.7578 3.267 ψMNX

-47.93 σφ

0.0528 0.0383

-0.0082 -0.0094

0.0083 0.0088

Mg(NO3)2 + MgSO4 + H2O

0.0447

-0.0046

0.0058

Cφ

unlikely. Consequently, the most dominating phenomenon which results from these interactions is the hydration of the ions. The interactions between ions and water molecules dominate. For these mixed electrolytes Mg2+/Cl-/NO3-/SO42-, the presence of MgSO4 in the ternary systems influences the interaction type in the solution [MgCl2 + MgSO4](aq). But for [Mg(NO3)2 + MgSO4](aq), the effect of ionic association in aqueous solutions of MgSO4 is dominated by the effect of Mg(NO3)2(aq) in aqueous solution, which is the hydration of ions by water molecules. 3.2 Osmotic Coefficient. Using the obtained experimental results of the water activity, the osmotic coefficients of water were evaluated for the ternary systems of magnesium Mg2+/Cl-/NO3-/SO42- at different ionic strength fractions (y) by

ref

0.0060 -0.0051 0.0232

23 24 8

4 this work this work

of magnesium sulfate in solution. This behavior can be explained by the presence of ionic association in the solution of magnesium sulfate. This phenomenon of ionic association was observed in aqueous binary solutions of sulfates.8 Indeed, many studies of the ion association in aqueous solutions in the MgSO4 system have been reported.9-12 In general, there are two types of ion association in aqueous electrolyte solutions. The first type is strong association of cations and anions by ionic or covalent bonds. The second type is the weak association complex in the form of solvent-separated ion pairs, associated by predominantly electrostatic forces. Zhang and Chan13 have observed, using Raman spectra, the characteristics of contact ion pairs between Mg2+ and SO42- and proposed a chain structure composed of the contact biden tate ion pairs in highly supersaturated MgSO4 solutions. But for the solute Mg(NO3)2(aq) in aqueous solutions, several studies14-19 with some techniques have concluded that contact ion pairs do not form up to saturation. These studies confirmed that NO3- does not bind directly to Mg2+ within the experimental concentration range and imply that complex formation between Mg2+ and NO3- in aqueous solutions is

φ)-

( ) 1000

Mw

∑

‚ln aw

(1)

Vimi

i

where νi is the number of ions released by dissociation; mi is the molality of solute i; and Mw is the molar mass of water. The obtained osmotic coefficients are listed in Tables 1 and 2 and are shown in Figure 6. The plots of osmotic coefficients of ternary systems of magnesium Mg2+/Cl-/NO3-/SO42- against the square root of total ionic strength I1/2 (Figure 6) are between those for the correspondent binary systems. The order is φ((Mg(NO3)2) or φ(MgCl2) > φ(y ) 0.80) > φ(y ) 0.50) > φ(y ) 0.20) > φ(MgSO4). We can observe that when we added the electrolyte MgSO4(aq) to a binary solution of Mg(NO3)2(aq) or MgCl2(aq) for [Mg(NO3)2 + MgSO4](aq) and [MgCl2 + MgSO4](aq), respectively, the solution approached the properties of the added salt MgSO4(aq). 3.3 ActiWity Coefficient. The osmotic coefficient of the mixed electrolyte solution of salts Mg2+/Cl-/ NO3-/SO42-with a

Table 4. Activity Coefficients γMgCl2 (γ1) and γMgSO4 (γ2) in {yMgCl2 + (1 - y)MgSO4}(aq) against Total Ionic Strength I for Different Ionic Strength Fractions (y) of MgCl2 y ) 0.00

y ) 0.20

I

y ) 0.50

I

y ) 0.80

I

y ) 1.00

I

I

mol‚kg-1

γ1

γ2

mol‚kg-1

γ1

γ2

mol‚kg-1

γ1

γ2

mol‚kg-1

γ1

γ2

mol‚kg-1

γ1

γ2

0.80 2.40 3.20 4.00 5.60 6.00 8.00 10.00 12.00

0.406 0.363 0.365 0.374 0.410 0.423 0.505 0.626 0.801

0.1440 0.0661 0.0576 0.0523 0.0466 0.0459 0.0449 0.0474 0.0530

2.25 3.00 3.75 4.50 6.00 7.50 9.00 10.50 11.10

0.390 0.398 0.415 0.438 0.505 0.602 0.735 0.915 1.001

0.0747 0.0670 0.0625 0.0600 0.0588 0.0612 0.0667 0.0754 0.0798

1.20 2.40 3.60 4.80 6.00 7.20 8.40 9.60 10.80 12.00

0.428 0.436 0.481 0.556 0.661 0.804 0.995 1.250 1.590 2.043

0.105 0.0834 0.0780 0.0793 0.0851 0.0949 0.109 0.129 0.155 0.190

3.00 4.01 4.90 6.00 6.95 8.12 9.00 9.92 10.86 12.00

0.518 0.620 0.716 0.866 1.048 1.369 1.703 2.157 2.749 3.671

0.0937 0.101 0.109 0.123 0.142 0.176 0.211 0.259 0.321 0.417

0.60 1.50 3.00 4.50 6.00 7.50 9.00 12.00

0.485 0.475 0.563 0.742 1.039 1.513 2.272 3.494

0.144 0.108 0.104 0.122 0.157 0.216 0.311 0.464

Table 5. Activity Coefficients γMg(NO3)2 (γ1) and γMgSO4 (γ2) in {yMg(NO3)2 + (1 - y)MgSO4}(aq) against Total Ionic Strength I at Different Ionic Strength Fractions (y) of Mg(NO3)2 y ) 0.00

y ) 0.20

I

y ) 0.50

I

y ) 0.80

I

y ) 1.00

I

I

mol‚kg-1

γ1

γ2

mol‚kg-1

γ1

γ2

mol‚kg-1

γ1

γ2

mol‚kg-1

γ1

γ2

mol‚kg-1

γ1

γ2

0.80 1.60 2.40 3.20 4.00 5.60 7.20 8.00 10.00 12.00

0.407 0.371 0.359 0.359 0.365 0.393 0.440 0.470 0.568 0.705

0.1140 0.0809 0.0661 0.0576 0.0523 0.0466 0.0448 0.0449 0.0474 0.0530

1.50 2.25 3.00 3.75 4.50 6.00 7.50 9.00 9.75 10.50

0.390 0.381 0.382 0.391 0.405 0.449 0.513 0.598 0.650 0.710

0.0879 0.0729 0.0644 0.0593 0.0561 0.0535 0.0544 0.0578 0.0605 0.0638

1.20 2.40 3.60 4.80 6.00 7.20 8.40 9.60

0.423 0.410 0.429 0.466 0.520 0.592 0.683 0.796

0.1030 0.0774 0.0681 0.0652 0.0660 0.0696 0.0757 0.0846

3.00 3.95 4.90 6.00 6.95 8.12 9.00 9.92 10.86 12.00

0.450 0.483 0.526 0.593 0.667 0.781 0.886 1.015 1.169 1.389

0.0780 0.0759 0.0760 0.0790 0.0844 0.0950 0.106 0.121 0.139 0.166

0.30 0.90 1.50 3.00 4.50 6.00 7.50 9.00 12.00 15.00

0.530 0.466 0.451 0.471 0.537 0.642 0.790 0.989 1.590 2.567

0.191 0.123 0.101 0.0817 0.0803 0.0878 0.103 0.126 0.210 0.378

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Journal of Chemical and Engineering Data, Vol. 52, No. 6, 2007

Figure 7. Activity coefficients γMgCl2 and γMgSO4 in {yMgCl2 + (1 - y)MgSO4}(aq) as a function of the square root of the ionic strength I1/2 at different ionic strength fractions (y) of MgCl2. O, y ) 0.00; 0, y ) 0.20; (, y ) 0.50; 2, y ) 0.80; +, y ) 1.00.

Mg2+

common cation is given by eq 2 based on the Pitzer ion interaction model20-22 φ-1)(

∑ m)

-1

i

i

∑ m )C

(

z

1/2

(zcza)

φ ca

]

+

{

2Ifφ + 2

c

c

∑∑ c

a

a

[

The activity coefficient γMX of MX in a common cation mixture is given by Pitzer’s model as

Bφca +

mcmc′[θcc′ + Iθ′cc′ +

c′

∑

a

a′

[θaa′ + Iθ′aa′ +

a′

∑mψ c

c

( )∑ [ ( ) ] ( )∑ [ ∑ VM

maψcc′a] +

a

∑∑mm a

∑∑mm

Figure 8. Activity coefficients γMg(NO3)2 and γMgSO4 in {yMg(NO3)2 + (1 - y)MgSO4}(aq) as a function of the square root of the ionic strength I1/2 at different ionic strength fractions (y) of Mg(NO3)2. O, y ) 0.00; 0, y ) 0.20; (, y ) 0.50; 2, y ) 0.80; +, y ) 1.00.

caa′]

}

ln γMX ) |zMzX|fγ + 2 VX

(2)

VM

c

where is a function of ionic strength expressing the effect of long-range electrostatic forces; I is the total ionic strength of the solution; θ and θ′ are binary mixing parameters involving unlike ions of the same sign; and ψ is a ternary mixing parameter for two different ions of the same sign with a third ion of the opposite sign. The second virial coefficient Bφ of the ions is defined as Bφ ) β(0) + β(1) exp(-RΙ1/2)

VX

θXa + 2

(3)

β(0), β(1), and Cφ are ion interaction parameters which are functions of temperature and pressure, where R ) 2 (mol‚kg-1)-1.

c

ma BMa + (

a

mc BcX + (

V

c

∑ ∑ m m {|z

fφ

V

MzX

a

1

∑∑mm 2 1 2

c

∑∑ a

mz)CcX +

Ma

+

() ] VM VX

θMc +

a′

[( ) | | ] [( ) | | ] VX

ψcc′X + zMzX θ′cc′ + V VM mama′ ψMaa′ + zMzX θ′aa′ (4) V c

c

z

|B′ca + V-1[2VMzMCca + VMψMca +

a

VXψcaX]} +

∑ m )C

c′

with f γ ) -Aφ

[

I1/2 2 + ln(1 + 1.2I1/2) 1.2 1 + 1.2I1/2

( )

]

(5)

Journal of Chemical and Engineering Data, Vol. 52, No. 6, 2007 2193

B′MX )

( )[ 2β(1) MX

( ) 2β(1) MX

BMX ) β(0) MX +

R2I2

[1 - (1 + RI1/2) exp(-RI1/2)]

1 -1 + 1 + RI1/2 + R2I exp(-RI1/2) 2

R2I2

(

CMX )

)

CφMX 2|zMzX|1/2

(6)

]

(7)

(8)

From the osmotic coefficients determined from the experimental water activities of the studied mixture at different ionic strength fractions, it is possible to determine the unknowns θMN and ψMNX. These parameters were used to predict the solute activity coefficients in the mixture. θMN and ψMNX are estimated by a graphical procedure. The quantity ∆φ is defined by the difference between the experimental values φexptl and that calculated φcalcd from eq 2. This yields

( ) ∑m

i

i

∆φ ) θMN + mXψMNX 2mMmN mX ) |zM/zX|mMX + |zN/zX|mNX

(9) (10)

A plot of ∆φ versus total molality mtot should give a straight line from which intercept θMN and slope ψMNX can be calculated. The corresponding values of ionic parameters β(0), β(1), and Cφ of the pure electrolytes MgCl2(aq), Mg(NO3)2(aq), and MgSO4(aq) were obtained from Pitzer’s expressions by a fit of the experimental osmotic coefficients given in our previous works8,23-24 and the unknown mixing ionic parameters θMN and ψMNX which are listed in Table 3. The standard deviations on φ over the entire range of total ionic strengths are between 0.0088 > σφ > 0.0058. The activity coefficients of MgCl2(aq), Mg(NO3)2(aq), and MgSO4(aq), in their ternary systems [MgCl2 + MgSO4](aq) and [Mg(NO3)2 +MgSO4](aq), listed in Tables 4 and 5, were calculated from eq 9 using the obtained ionic mixing parameters. The plots of activity coefficients in the mixtures against the square root of the ionic strength I1/2 are shown in Figures 7 to 9. From the theoretical point of view, it is suitable to compare and explain the behavior of various ionic solutes, magnesium, in wide ranges of composition. Activity coefficients in the various mixtures {Mg2+/Cl-/ NO3-/SO42-}(aq) as a function at molality show that the behavior for each aqueous solution is similar with that of its corresponding binary solution. It is also noted that these curves for systems [Mg(NO3)2 + MgSO4](aq) and [MgCl2 + MgSO4](aq) are between those of their binary correspondents, but for [MgCl2 + Mg(NO3)2](aq), the behavior of the solute activity coefficients is identical, and we can conclude that the structure of solution is not perturbed by addition of the electrolyte solute MgCl2 to Mg(NO3)2(aq) in all the ranges of composition. The activity coefficient γMgSO4 in their systems has lower values (1 > γ). These values confirmed the presence of the phenomenon of ionic association manifested in these systems, and contrarily for γMg(NO3)2 or γMgCl2, the higher values than unity reflect the interaction in the solution (Table 6).

Figure 9. Activity coefficients γMgCl2 and γMg(NO3)2 in {yMgCl2 + (1 y)Mg(NO3)2}(aq)4 as a function of the square root of the ionic strength I1/2 at different ionic strength fractions (y) of MgCl2. O, y ) 0.00; 0, y ) 0.20; (, y ) 0.50; 2, y ) 0.80; +, y ) 1.00. Table 6. Range of Activity Coefficients in Their Systems system

γ1

γ2

[Mg(NO3)2 + MgSO4](aq) 2.567> γMg(NO3)2 > 0.359 0.378 > γMgSO4 > 0.046 [MgCl2 + MgSO4](aq) 3.670 > γMgCl2 > 0.363 0.464 > γMgSO4 > 0.046 [MgCl2 + Mg(NO3)2](aq) 55 > γMgCl2 > 0.479 38 > γMg(NO3)2 > 0.471

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Received for review April 25, 2007. Accepted August 18, 2007.

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